The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/(2N)1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secul...

Let G be a graph of order n with (0, 1)-adjacency matrix A. An eigenvalue σ of A is said to be an eigenvalue of G, and σ is a main eigenvalue if the eigenspace EA(σ) is not orthogonal to the all-1 vector in IR. Always the largest eigenvalue, or index, of G is a main eigenvalue, and it is the only main eigenvalue if and only if G is regular. We say that G is an integral graph if every eigenvalue...

The Lanczos method is often used to solve a large and sparse symmetric matrix eigenvalue problem. It is well-known that the single-vector Lanczos method can only find one copy of any multiple eigenvalue. To compute all or some of the copies of a multiple eigenvalue, one has to use the block Lanczos method which is also known to compute clustered eigenvalues much faster than the single-vector La...

The low-rank damping term appears commonly in quadratic eigenvalue problems arising from physical simulations. To exploit the low-rank damping property, we propose a Padé approximate linearization (PAL) algorithm. The advantage of the PAL algorithm is that the dimension of the resulting linear eigenvalue problem is only nC `m, which is generally substantially smaller than the dimension 2n of th...

We consider polynomial eigenvalue problems P(A; ;)x = 0 in which the matrix polynomial is homogeneous in the eigenvalue (;) 2 C 2. In this framework innnite eigenvalues are on the same footing as nite eigenvalues. We view the problem in projective spaces to avoid normalization of the eigenpairs. We show that a polynomial eigenvalue problem is well-posed when its eigenvalues are simple. We deene...

In this article, we construct aC0 linear finite element method for two fourth-order eigenvalue problems: the biharmonic and the transmission eigenvalue problems. The basic idea of our construction is to use gradient recovery operator to compute the higher-order derivatives of a C0 piecewise linear function, which do not exist in the classical sense. For the biharmonic eigenvalue problem, the op...

We consider polynomial eigenvalue problems P(A, α, β)x = 0 in which the matrix polynomial is homogeneous in the eigenvalue (α, β) ∈ C2. In this framework infinite eigenvalues are on the same footing as finite eigenvalues. We view the problem in projective spaces to avoid normalization of the eigenpairs. We show that a polynomial eigenvalue problem is wellposed when its eigenvalues are simple. W...

In this paper we propose numerical algorithms for solving large-scale quadratic eigenvalue problems for which a set of eigenvalues closest to a fixed target and the associated eigenvectors are of interest. The desired eigenvalues are usually with smallest modulo in the spectrum. The algorithm based on the quadratic Jacobi-Davidson (QJD) algorithm is proposed to find the first smallest eigenvalu...

We present Gerschgorin-type eigenvalue inclusion sets applicable to generalized eigenvalue problems. Our sets are defined by circles in the complex plane in the standard Euclidean metric, and are easier to compute than known similar results. As one application we use our results to provide a forward error analysis for a computed eigenvalue of a diagonalizable pencil.

This paper concerns with the sensitivity analysis for the multivariate eigenvalue problem (MEP). The concept of a simple multivariate eigenvalue of a matrix is generalized to the MEP and the first-order perturbation expansions of a simple multivariate eigenvalue and the corresponding multivariate eigenvector are presented. The explicit expressions of condition numbers, perturbation upper bounds...